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arXiv:2603.05335v2 Announce Type: replace Abstract: Four distinct admissibility geometries govern sequential and distribution-free inference: Blackwell risk dominance over convex risk sets, anytime-valid admissibility within the nonnegative supermartingale cone, marginal coverage validity over exchangeable prediction sets, and Ces`aro approachability (CAA) admissibility, which reaches the risk-set boundary via approachability-style arguments rather than explicit priors. We prove a criterion separation theorem: the four classes of admissible procedures are pairwise non-nested. Each geometry carries a different certificate of optimality: a supporting-hyperplane prior […]

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arXiv:2511.20888v2 Announce Type: replace Abstract: This paper argues that DNNs implement a computational Occam’s razor — finding the `simplest’ algorithm that fits the data — and that this could explain their incredible and wide-ranging success over more traditional statistical methods. We start with the discovery that the set of real-valued function $f$ that can be $epsilon$-approximated with a binary circuit of size at most $cepsilon^{-gamma}$ becomes convex in the `Harder than Monte Carlo’ (HTMC) regime, when $gamma>2$, allowing […]

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arXiv:2511.09500v4 Announce Type: replace Abstract: We study the problem of denoising when only the noise level is known, not the noise distribution. Independent noise $Z$ corrupts a signal $X$, yielding the observation $Y = X + sigma Z$ with known $sigma in (0,1)$. We propose emph{universal} denoisers, agnostic to both signal and noise distributions, that recover the signal distribution $P_X$ from $P_Y$. When the focus is on distributional recovery of $P_X$ rather than on individual realizations of $X$, […]

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arXiv:2502.10328v4 Announce Type: replace Abstract: Markov Chain Monte Carlo (MCMC) algorithms are essential tools in computational statistics for sampling from unnormalised probability distributions, but can be fragile when targeting high-dimensional, multimodal, or complex target distributions. Parallel Tempering (PT) enhances MCMC’s sample efficiency through annealing and parallel computation, propagating samples from tractable reference distributions to intractable targets via state swapping across interpolating distributions. The effectiveness of PT is limited by the often minimal overlap between adjacent distributions in challenging […]

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arXiv:2502.02861v4 Announce Type: replace Abstract: The field of algorithms with predictions incorporates machine learning advice in the design of online algorithms to improve real-world performance. A central consideration is the extent to which predictions can be trusted — while existing approaches often require users to specify an aggregate trust level, modern machine learning models can provide estimates of prediction-level uncertainty. In this paper, we propose calibration as a principled and practical tool to bridge this gap, demonstrating the […]

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arXiv:2405.17573v3 Announce Type: replace Abstract: We study Leaky ResNets, which interpolate between ResNets and Fully-Connected nets depending on an ‘effective depth’ hyper-parameter $tilde{L}$. In the infinite depth limit, we study ‘representation geodesics’ $A_{p}$: continuous paths in representation space (similar to NeuralODEs) from input $p=0$ to output $p=1$ that minimize the parameter norm of the network. We give a Lagrangian and Hamiltonian reformulation, which highlight the importance of two terms: a kinetic energy which favors small layer derivatives $partial_{p}A_{p}$ […]

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arXiv:2603.24594v1 Announce Type: cross Abstract: We introduce the Multilevel Euler-Maruyama (ML-EM) method compute solutions of SDEs and ODEs using a range of approximators $f^1,dots,f^k$ to the drift $f$ with increasing accuracy and computational cost, only requiring a few evaluations of the most accurate $f^k$ and many evaluations of the less costly $f^1,dots,f^{k-1}$. If the drift lies in the so-called Harder than Monte Carlo (HTMC) regime, i.e. it requires $epsilon^{-gamma}$ compute to be $epsilon$-approximated for some $gamma>2$, then ML-EM […]

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arXiv:2603.24545v1 Announce Type: cross Abstract: We study the problem of detecting local geometry in random graphs. We introduce a model $mathcal{G}(n, p, d, k)$, where a hidden community of average size $k$ has edges drawn as a random geometric graph on $mathbb{S}^{d-1}$, while all remaining edges follow the ErdH{o}s–R’enyi model $mathcal{G}(n, p)$. The random geometric graph is generated by thresholding inner products of latent vectors on $mathbb{S}^{d-1}$, with each edge having marginal probability equal to $p$. This implies […]

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arXiv:2603.24495v1 Announce Type: cross Abstract: While the mathematical foundations of score-based generative models are increasingly well understood for unconstrained Euclidean spaces, many practical applications involve data restricted to bounded domains. This paper provides a statistical analysis of reflected diffusion models on the hypercube $[0,1]^D$ for target distributions supported on $d$-dimensional linear subspaces. A primary challenge in this setting is the absence of Gaussian transition kernels, which play a central role in standard theory in $mathbb{R}^D$. By employing an […]

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arXiv:2603.24384v1 Announce Type: cross Abstract: The theory of Local Intrinsic Dimensionality (LID) has become a valuable tool for characterizing local complexity within and across data manifolds, supporting a range of data mining and machine learning tasks. Accurate LID estimation requires samples drawn from small neighborhoods around each query to avoid biases from nonlocal effects and potential manifold mixing, yet limited data within such neighborhoods tends to cause high estimation variance. As a variance reduction strategy, we propose an […]

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